Abstract: By the kernel of a one-sided (left or right) congruence on an inverse semigroup , we mean the set of -classes which contain idempotents of . We provide a set of independent axioms characterizing the kernel of a one-sided congruence on an inverse semigroup and show how to reconstruct the one-sided congruence from its kernel. Next we show how to characterize those partitions of the idempotents of an inverse semigroup which are induced by a one-sided congruence on and provide a characterization of the maximum and minimum one-sided congruences on inducing a given such partition. The final two sections are devoted to a study of indempotent-separating one-sided congruences and a characterization of all inverse semigroups with only trivial full inverse subsemigroups. A Green-Lagrange-type theorem for finite inverse semigroups is discussed in the fourth section.